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to allow for the temperature dependence of g and to calculate g values in solutions Section 10.8

of several electrolytes. Ionic Association
In the 1970s, Pitzer and coworkers developed equations for calculating g values

in concentrated aqueous electrolyte solutions [Pitzer (1995), chaps. 17 and 18; Pitzer
(1991), chap. 3]. Although Pitzer’s approach is based on a statistical-mechanical the-
ory of interactions between ions in the solution, his equations have a substantial dose
of empiricism, in that the mathematical forms of some of the terms in the equations
were chosen by seeing what forms give the best fit to data. Moreover, the equations
contain parameters whose values are not calculated theoretically but are chosen to fit
activity-coefficient or osmotic-coefficient data for the electrolyte(s) in question.
The Pitzer equations are widely used and have been applied to study reaction and
solubility equilibria in such systems as seawater, the Dead Sea, lakes, oil-field brines,
and acidic mine-drainage waters with excellent results. The value of the Pitzer equa-
tions is for dealing with solutions of several electrolytes, where their performance is
usually better than the Meissner model (J. F. Zemaitis et al., Handbook of Aqueous
Electrolytes, Design Institute for Physical Property Data, 1986).
For multicomponent electrolyte solutions with ionic strengths above 10 or
15 mol/kg, the Pitzer equations usually do not apply. Such high ionic strengths occur
in atmospheric aerosols, such as sea-spray aerosols, where evaporation of water pro-
duces solutions supersaturated in NaCl. (The flux of sea salt between the oceans and
the atmosphere is estimated at 10 15 g per year.) Pitzer and coworkers developed a
version of the Pitzer equations that is based on mole fractions instead of molalities and
that applies at extremely high concentrations [see Pitzer (1995), pp. 308–316].

10.8 IONIC ASSOCIATION

In Section 10.5, a strong electrolyte in aqueous solution was assumed to exist entirely in
the form of ions. Actually, this picture is incorrect, and (except for 1:1 electrolytes) there
is a significant amount of association between oppositely charged ions in solution to
yield ion pairs. For a true electrolyte, we start out with ions in the crystal, get solvated
ions in solution as the crystal dissolves, and then get some degree of association of sol-
vated ions to form ion pairs in solution. The equilibrium for ion-pair formation is

M 1sln2 X 1sln2 ∆ MX z z 1sln2 (10.69)
z
z

2
For example, in an aqueous (aq) Ca(NO ) solution, (10.69) reads Ca (aq) NO (aq)
3
3 2

∆ Ca(NO ) (aq).
3
The concept of ion pairs was introduced in 1926 by Bjerrum (Bockris and Reddy,
vol. I, sec. 3.8; Davies, chap. 15). Bjerrum proposed (rather arbitrarily) that two op-
positely charged ions close enough to make the potential energy of attraction between
them larger in magnitude than 2kT [where k is Boltzmann’s constant (3.57)] be con-
sidered an ion pair. Bjerrum used a model similar to that of Debye and Hückel to find
a theoretical expression for the degree of association to ion pairs as a function of the
electrolyte concentration, z , z , T, e , and the mean ionic diameter a. His theory in-

r
dicated that in water, association to ion pairs is usually negligible for 1:1 electrolytes
but can be quite substantial for electrolytes with higher z z values, even at low con-

centrations. As z and z increase, the magnitude of the interionic electrostatic

attraction increases and ion pairing increases.
The solvent H O has a high dielectric constant (because of the polarity of the
2
water molecule). In solvents with lower values of e , the magnitude of the electrosta-
r
tic attraction energy is greater than in aqueous solutions. Hence, ion-pair formation in
these solvents is greater than in water. Even for 1:1 electrolytes, ion-pair formation is
important in solvents with low dielectric constants.